Split tube flexure

ABSTRACT

This application describes the design of the split-tube flexure, a unique precision revolute joint that exhibits a considerably larger range of motion and significantly better multi-axis revolute joint characteristics than a conventional flexure. The development of this joint enables the implementation of spatially-loaded revolute joint-based precision machines with well-behaved kinematic and dynamic characteristics and without the backlash and stick-slip behavior that would otherwise prevent precision machine control.

TECHNICAL FIELD

This invention relates generally to the field of mechanical connectionsand more particularly to flexure joints.

BACKGROUND ART

Effective implementation of precision control is largely influenced bythe open-loop behavior of a machine. In particular, the presence of hardnonlinearities such as backlash and Coulomb friction results insignificant deterioration of machine control. The elimination of hardnonlinearities enables effective and accurate position and force controlof a precision machine.

Some classes of machines are particularly sensitive to the presence ofbacklash and Coulomb friction. Due to the physics of scaling, devicesthat operate on a microscopic scale are influenced by highly nonlinearsurface forces to a much greater degree than those of a conventionalscale. Consequently, a scaled-down micromachine is significantly moresensitive to Coulomb friction than its conventional-scale counterpart.Successful development of small-scale and micro-scale precision machinesrequires elimination or intelligent minimization of surface forcebehavior.

Machines that operate in a zero-gravity environment, such as satellitesand spacecraft, are also particularly sensitive to Coulomb-type bearingfriction, primarily because gravity is no longer the dominate mechanicalinfluence.

Flexure-based Design

Most conventional mechanisms rely on sliding and rolling at afundamental level. Kinematic linkages, for example, cannot beconstructed without revolute joints, which almost universallyincorporate roller or journal bearings. In conventional machines, suchdesigns can provide low friction rotation while bearing significantloads. Precision motion, however, as well as small-scale and zerogravity applications, require open-loop behavior that is devoid of anysignificant stick-slip behavior. A flexure-base joint, which utilizesdeformation as a means of providing movement, is a viable alternative tothe conventional revolute joint that does not exhibit any significantbacklash or Coulomb friction and is free of lubricants. A diagram of aconventional flexure 2 is shown in FIG. 1. The basic characteristics ofconventional flexure joints have been studied by several researchers[1,2,3,4].

References

[1] Goldfarb, M. and Celanovic, N., “Minimum Surface-Effect MicrogripperDesign for Force-Reflective Telemanipulation of a MicroscopicEnvironment,” Proceedings of the ASME International MechanicalEngineering Conference and Exposition, November 1996.

[2] Horie M., Nozaki T., Ikegami K., and Kobayashi, F., “Design Systemof Super Elastic Hinges and its Application to Manipulator forMicro-Bonding by Adhesives,” Proceedings of the International Symposiumon Microsystems, Intelligent Materials, and Robots, pp. 185-188, 1995.

[3] Paros, J. and Weisbord, L., “How to Design Flexure Hinges,” MachineDesign, Vol. 37, No. 27, pp. 151-156, 1965.

[4] Ragulskis K., Arutunian M., Kochikian A., and Pogosian M., “A Studyof Fillet Type Flexure Hinges and Their Optimal Design,” VibrationEngineering, pp. 447-452, 1989.

If properly designed, a flexure-based structure can approximate themotion of a complex kinematic linkage with negligible stick-slipfriction and no backlash. Additionally, the absence of rolling andsliding surfaces produces a device that is free of lubricants and thusextremely conducive to clean environments. Conventional flexures,however have several significant deficiencies. One particularlyrestrictive deficiency is the limited range of motion. Depending on theflexure geometry and material properties, a flexure will beginexhibiting plastic deformation at ranges on the order of five to tendegrees of rotation. In contrast, an ideal revolute joint has aninfinite range of motion.

Another significant problem with conventional flexures is the poorproperties exhibited when subjected to multi-axis loading. An idealrevolute joint is infinitely rigid in all directions of loading exceptthe desired axis of rotation. In contrast, a conventional flexureexhibits a significant stiffness along the desired axis of rotation andsignificant compliance along all other axes of loading. A flexure-basedjoint, for example, will exhibit twist-bend buckling when subjected totwisting. The multi-axis behavior of conventional flexure joints resultsin both kinematic and dynamic problems, especially when utilizingnon-collocated control that relies upon kinematic transformations fortask-space accuracy.

What is needed, then, is a joint that enables the implementation ofprecision spatially-loaded revolute joint-based machines withwell-behaved kinematic characteristics and without the backlash andstick-slip behavior that would otherwise prevent precision control.

The new joint or flexure should: exhibit no backlash or stick-slipbehavior; exhibit off-axis stiffnesses significantly greater than acomparable conventional flexure; enable greater range of motion than acomparable conventional flexure; and withstand more load than aconventional flexure. Such a joint is lacking in the prior art.

DISCLOSURE OF THE INVENTION

As mentioned previously, an ideal revolute joint is characterized byzero stiffness along the axis of rotation and infinite stiffness alongall other axes of loading. Conventional flexure joints offer the benefitof zero backlash and Coulomb friction, but not without limitation.Conventional flexure joints are constrained to a small range of motionand are subject to significant stiffness along the axis of rotation andsignificant compliance along other axes. This application describes anew flexure that exhibits a considerably larger range of motion andsignificantly better multi-axis revolute joint characteristics than aconventional flexure.

The design of the joint is based upon contrasting the torsionalcompliance of an open section with its stiffnesses in compression andbending. The torsional mechanics of closed section and open sectionmembers are fundamentally and significantly different, while the bendingand compressive mechanics of the members are quite similar. Thisdifference in mechanics enables minimization of torsional stiffness andmaximization of all other stiffnesses in a nearly decoupled manner.

One embodiment of the invention includes a split-tube flexure. Thesplit-tube flexure includes a thin-waled shaft, a first arm, and asecond arm. The shaft is not required to be thin-walled. It may be ageneric tube having nearly any conventional of cross-section, includingpolygonal. Preferably, the tube is a thin-walled nearly closed splitcylinder, or hollow shaft. The wall thickness is variable as well. Thus,this invention is not limited to thin-walled hollow shafts. Designrequirements, such as a required strength or stiffness, will drive thedesign parameters (the structural dimensions and material properties).

The thin-walled shaft includes a wall, first and second ends, and alength between the first and second ends. The wall defines a lengthwiseslit therein. The first link is attached to the shaft. The second linkincludes a length and is attached to the shaft transversely to the slit.In one embodiment, the shaft includes mounting holes at each end-toattach the first and second links. The mounting holes should be on alongitudinal axis opposite the slit.

Another embodiment of the split-tube flexure includes a secondthin-walled shaft attached to the single split-tube flexure to form acompound split-tube flexure. Typically the shafts are aligned co-linear(end-to-end), though the only requirement is that the mounting holes ofboth tubes remain co-linear. The entire cross section of the first endof the first tube need not be co-linear with the entire cross section ofthe first end of the second tube, i.e. the respective centroids need notbe aligned. A first end of the first tube typically faces a first end ofthe second tube. The first link is attached to the first ends of thetubes. A second link is attached to second ends of the tube. The linksneed not be attached at end points of course.

The invention includes a method of providing pivotal motion with aflexure joint. The method typically entails providing a firstthin-walled shaft (or similar design constrained structure) having alengthwise slit; attaching a first link to the thin-walled shaft, wherethe first link extends transversely to the slit (typicallyperpendicular). And attaching a second link to the first thin-walledshaft, the second link also extending transverse to the slit. Applying aforce at the first arm, where the force has a vector componentperpendicular to the slit. Applying the axial force will create reactiveforces at the second arm, and thereby achieve minimal force resistancealong the axis of rotation and relatively large force resistance alongother axes of the flexure joint.

One objective of the invention is to provide a new flexure-basedrevolute joint that offers significantly better properties than aconventional flexure. More specific objectives of the invention are toprovide a joint which exhibits little to no backlash or stick-slipbehavior, and which exhibits off-axis stiffnesses significantly greaterthan a comparable conventional flexure. A further objective of theinvention is to provide a mechanism which enables greater range ofmotion compared to a conventional flexure.

Another objective of the invention is to provide a joint whichwithstands more load than a conventional flexure.

A broader objective of the invention is to provide a joint that enablesthe implementation of a precision spatially-loaded revolute joint-basedmachines with well-behaved kinematic characteristics and without thebacklash and stick-slip behavior that would otherwise prevent precisioncontrol. Another object of the invention is to provide this flexure in alubricant free environment suitable for ‘clean’ environments andspace-based applications.

FIG. 1 is a perspective view of a conventional flexure joint, indicatingthe nominal joint axis of rotation.

FIGS. 2a and 2 b are perspective views of closed (left—FIG. 2a) and open(right FIG. 2b) section hollow shafts.

FIG. 3a, b, and c are perspective views of the split-tube flexure of thepresent invention in relaxed and flexed positions.

FIG. 4 is a perspective of view showing the axes of loading for asplit-tube flexure joint of the present invention.

FIG. 5 is a perspective view showing the axes of loading for aconventional flexure joint.

FIGS. 6a and 6 b are perspective views showing the geometry of asplit-tube flexure joint of the present invention.

FIG. 7 is a perspective view showing the geometry of a conventionalflexure joint.

FIG. 8a and 8 b are perspective (solid model) views of anotherembodiment of the split-tube flexure of the present invention as used ina two degree-of-freedom five-bar parallel linkage.

FIG. 9 is a side view of the two dof linkage shown in

FIG. 8 incorporated into a microbot.

FIG. 10 is an enlarged- side view of the two dof linkage shown in FIG.9.

FIG. 11 is a pin-joint representation of the two dof linkage shown inFIG. 10.

FIGS. 12a and 12 b are perspective views illustrating the compound (top)and the simple (bottom) embodiments of the split-tube flexure of thepresent invention.

BEST MODE FOR CARRYING OUT THE INVENTION The Mechanics of Open VersusClosed Sections

FIG. 2 shows two hollow shafts, 4 and 6, that are in every manneridentical except that one, 6, has a slit along its long axis (alengthwise slit, or slit). Each shaft having a length L, an outer radiusR, and a wall thickness t. Though geometrically similar, the mechanicsof how each bears a torsional load are quite different. For purposes oftorsional mechanics, the wholly intact shaft reacts mechanically:in thesame mode as a solid shaft, while the slitted shaft behaves mechanicallyas a thin flat plate. (Note that in the limit as R approaches infinity,a split cylinder approaches a flat plate.) This dissimilarity inbehavior results in very different torsional stiffnesses. Definingtorsional stiffness as the ratio of torque about the long axis to theangular deflection about the same, a stress analysis (assuming typicalproperties such as linearly elastic, homogeneous, isotropic material)illustrates the differences in torsional mechanics.

The torsional stiffness of the closed section, k_(cs), is given by:$k_{cs} = {\frac{2\quad \pi \quad G}{L}R^{3}t}$

and that of the open section, k_(os), given by:$k_{os} = {\frac{2\quad \pi \quad G}{3L}R\quad t^{3}}$

where G is the shear modulus of elasticity, and L, R, and t are thelength, outside radius, and wall thickness of the shaft, respectively.Since by definition, the geometry of a thin-walled shaft is such thatt<<R (typically t less than 0.15 R), the torsional stiffness of the opensection is significantly less than of its closed counterpart. Forexample, a closed-section shaft of radius R=2.5 mm and wall thicknesst=0.05 mm would exhibit a torsional stiffness approximately 7500 timesthat of a geometrically similar open section.

Though the torsional mechanics of the open and closed section shafts arequite different, the mechanics of bending are in essence the same. Bothopen and closed section shafts have a bending stiffness given by:$k_{b} = {\frac{\pi \quad E}{L}R^{3}t}$

where E is the modulus of elasticity, and the other variables are asdefined previously. Note that this stiffness is quite similar to thetorsional stiffness of a closed section shaft. In fact, in the limit oft<<R, the joint structure would have a zero torsional stiffness and aninfinite bending stiffness, emulating the ideal revolute joint withoutthe corresponding backlash and Coulomb friction. This, in essence,characterizes the design of the split-tube flexure. FIG. 3a-3 c show asplit-tube flexure hinge. FIG. 3a is of a relaxed position, 3 a and 3 care of flexed positions, i.e. the links have been moved through someangle of revolution. Note that the axis of rotation is along the top ofthe tube opposite the split, and not through the center of the tube.Also, the links adjoined by the flexure hinge remain parallel andperpendicular to the axis of rotation though the slit moves with torsionof the cylinder.

The Split-tube Versus the Conventional Flexure

FIG. 6a and 6 b show an embodiment of this invention, a singlesplit-tube flexure 10. It shows a thin-walled shaft 20 including a slit30. The thin-walled shaft 20 generally includes an outer radialdimension R, a wall 25 having, thickness t, and a length L. The length Lbeing defined between mounting holes near a first end 21 and :a secondend 23. The slit 30 is a lengthwise slit typically running the length ofthe shaft 20. In the embodiment shown, the shaft 20 includes acylindrical, or tube, shape. Also included in the split-tube flexure 10is a first link 40 attached to the shaft wall 25, and a second link 50attached to the shaft wall 25. The first and second links 40 and 50 areattached to the wall at the first and second longitudinally spacedpositions along the length of the tube 20. The torsional displacement islinearly related to the longitudinal spacing. The longitudinal spacingshould be such that the links are not end to end, or the resultantflexure motion will be from bending of the wall, not torsion. It will beapparent that attaching the links on the wall includes attaching thelinks on the wall edges. Also the links need not be straight rods orbars as shown in FIGS. 6a and 6 b.

The first link 40 and the second link 50 are shown with ends attached toends of the shaft 20 such that the links 40 and 50 are transverse to,and as.shown perpendicular to, the lengthwise slit 30. In alternateembodiments, the links 40 and 50 need not be perpendicular to the slit30, although, at least one link should be transverse to the slit 30.Transverse being defined as other than parallel. Also, the contactlocations will typically be in a line parallel to the slit 30,preferably opposite the slit 30. The contact locations need not bealigned parallel to the slit 30, but the resultant forces, axis ofrotation and the mathematics describing the structural dynamics willchange.

FIG. 6b shows the split-tube flexure 10 of FIG. 6a further annotated.The thin-walled shaft 20 includes a bottom side 22, a top side 24, afore side 26, and a back side 28. The first link 40 includes a firstportion 42, a second portion 44, and a length 43. The second link 50includes a first portion 52, a second portion 54, and a length 53. Thelinks 40 and 50 are shown attached to the topside 24 of the shaft 20.The links 40 and 50, though typically attached to the topside 24 of theshaft 20, need not be attached to it.

FIG. 7 shows a conventional flexure joint 2. It includes a flexure beamwidth b, a flexure member height h, and a flexure length 1.

An analytical comparison is useful in assessing the relative mechanicalcharacteristics of both a split-tube and a conventional flexure. FIGS. 4and 5 illustrate split-tube 10 and conventional flexure 2 geometry,respectively, and define the loads (axial loads F₁ and F₂, and torsionalloads M₁, M₂, and τ) and deflections (x, y, φ₁, φ₂, and θ) from whichrelevant stiffnesses can be, determined.

As previously mentioned, the objective in the design of a flexurerevolute joint is to achieve a minimal stiffness along the axis ofrotation and relatively large stiffnesses along all other axes. Theinventors refer to the stiffness along the revolute axis as the revolutestiffness, which is defined by k_(τ)=τ/θ. The off-axis stiffnesses ofprimary interest are the bending stiffnesses, defined by k_(b1)=M₁/θ₁and k_(b2)=M₂/θ₂ and the axial stiffnesses, defined by k_(a1)=F₁/x andk_(a2)=F₂/y. Also of primary interest when characterizing jointperformance is the allowable range of motion afforded by the joint, acharacteristic determined by material yield.

For purposes of comparison, a split-tube 10 and a conventional 2 flexurewere designed according to desired specifications that were determinedby the desired machine performance characteristics. The two Joints weredesigned to have the same revolute stiffness k_(τ), and the joints wererequired to have the same axial stiffness k_(a1) and to withstand agiven minimum axial load F₁. Additionally, the two joints were designedof the same stainless steel alloy. The resulting dimensions, as definedin FIGS. 6a and 7, are R=2.4 mm, t=0.05 mm, and L=9.5 mm for thesplit-tube flexure, and h=0.2 mm, b=0.2 mm, and l=5.4 mm for theconventional flexure.

The resulting stiffnesses, along with the ranges of motion are given inTable 1. The characteristics for the split-tube flexure 10 wereexperimentally verified. Experiments were additionally incorporated todetermine maximum load before buckling, a mechanical quantity which isnot as well analytically characterized as stiffness or yield. Theseexperiments indicated that the split-tube flexures 10 could withstandapproximately 8 Newtons before buckling, which is more than 3 times the2.4 Newton capability of the conventional flexure 2.

Table 2 incorporates the same information as Table 1, but representedinstead as the relative characteristics of the split-tube flexure withrespect to the conventional flexure. The ratios of k_(τ) and k_(a1) ofthe two flexures were set equal by design. One can observe from Table 2that all other off-axis stiffnesses of the split-tube flexure 10,k_(b1), k_(b2), and k_(a2) are three to four orders of magnitude largerthan the equivalent off-axis stiffnesses of the conventional flexure 2.Note also that the split-tube flexure 10 enables 150 degrees of motion,more than five times that of the equivalent conventional flexure 2. Thiscomparison clearly illustrates the improved revolute joint propertiesoffered by the split-tube flexure 10.

Another significant difference between the split-tube 10 andconventional 2 flexure involves the kinematic behavior of the joints. Anideal revolute joint provides a fixed axis of rotation so that themotion of one link with respect to the adjoining link can becharacterized as a pure rotation. The illustration of FIG. 1 depicts theinstantaneous revolute axis associated with a conventional flexurejoint. This axis does not remain fixed with respect to either link, butrather translates in the plane as the links rotate. In contrast, thesplit-tube flexure 10 has a fixed axis of rotation (opposite andparallel to the split) when the links are attached to the top side ofthe shaft and are perpendicular to the slit. This enableswell-characterized kinematics with a minimal set of measurements.

TABLE 1 Comparison of conventional and split-tube flexure revolute jointproperties. CONVENTIONAL SPLIT-TUBE PROPERTY FLEXURE FLEXURE k_(r)0.00525 0.00525 Nm/rad Nm/rad k_(b1) 0.00525 43.8 Nm/rad Nm/rad k_(b2)0.00806 43.8 Nm/rad Nm/rad k_(a1) 1.53 × 10⁶ N/m 1.45 × 10⁶ N/m k_(a2)540 N/m 1.45 × 10⁶ N/m θ_(max) ±14.7 degrees ±77.9 degrees

TABLE 2 Relative characteristics of the split-tube flexure with respectto the conventional flexure. CHARACTERISTICS OF SPLIT- TUBE RELATIVE TOPROPERTY CONVENTIONAL FLEXURE k_(r) 1 k_(b1) 8330 k_(b2) 5440 k_(a1)0.95 k_(a2) 2680 θ_(max) 5.3

Alternative embodiments will be apparent to those skilled the mechanicalstructures and dynamics arts. It will be apparent that the shaft wouldtypically be a thin-walled cylinder where t <<R, however, it is equallyapparent that the cross section of the shaft need not be circular. Theshaft 20 may include an alternative cross section such as a polygon orother multi-sided cross section.

Furthermore, design characteristics may dictate that the shaft (or tube)not be thin-walled. The desired stiffness ratio drives the relationshipbetween t and R. The invention is intended to include other thanthin-walled embodiments. A preferred embodiment would, however,typically use a thin-walled cylinder. The reactive forces correspondingto applied loads will, of course, change with various embodiments of theinvention. The invention can be used to achieve minimal force resistancealong (or about) an axis of rotation, and relatively large forceresistance along other axes. Linkages can be used to create minimalresistance along additional axes of rotation.

Mechanism Design

A revolute joint may be formed from a single or double split-tubearrangement. The double split-tube revolute joint, also referred to as acompound split-tube (see FIG. 12) has been incorporated into a parallellinkage to demonstrate kinematic features of the split-tube flexure.FIGS. 8a and 8 b show perspective view of a two degree-of-freedom (dof)parallel linkage 150 to locate a point 151 within a 2 dof space. FIG. 9shows a side view of the two dof linkage 150 incorporated into amicrobot 170. FIG. 10 shows an enlarged side view of the linkage 150shown in FIG. 9. As shown in FIGS. 8 through 10, a first set ofsplit-tubes 153 connects a first link 152 to a frame 162. A second setof split-tubes 155 connects a second link 154 to the frame 162. Thesecond link 154 being connected to a third link 156 with a third set ofsplit-tubes 157. A fourth link 158 is connected to the third link 156with a fourth set of split-tubes 159. A fifth set of split-tubes 161connects the first link 152 to the fourth link 158 between the point 151and the third link 156 connection 159.

Push-pull mechanisms 163 and 164 include a pivot at each end and a cableto attach the pivots to the links and an apparatus to push-pull thepush-pull mechanisms. The push-pull mechanisms will also be referred togenerally as knife-edge pivots. Knife-edge pivots 163 and 164 areconnected to the first link 152 and 154, respectively. Push-pulling theknife-edge pivots 163 and 164 in a vertical plane results in ahorizontal and vertical movement of point 151 through the kinematics ofthe linkage 150.

The location of point 151 can be determined as a function of link anglesθ₁ and θ₂. The link angles are related to the vertical distance theknife-edge pivots move through.

FIG. 11 shows a pin-joint representation of the linkage 150. Link 152 ispin connected at the same location as link 154. The common axis ofrevolution is achieved by aligning the split-tube 153 and 155 so thatthe side opposite the respective slits, i.e. the attachment points, areco-linear.

The microbot 170 shown in FIG. 9 includes a T-beam support 165 attachedto a base 172. An I-beam 173 attached to the base 172 supports supportflexures 174. A threaded rod 168 is supported by the support flexures174 and moved vertically by a voice coil 175. The threaded rod 168push-pulls the knife-edge pivot 163. A tensioning screw 166 is used totighten a cable 176 to secure the knife-edge pivot 163 to the secondlink 154 and a tension block 166. Similar connections are on the otherside of the microbot (not shown).

The parallel linkage used in the microbot is simply for illustrativepurposes. Clearly many other designs incorporating split-tube flexurejoints will also be apparent to those skilled in the arts. In theembodiments shown in FIGS. 8 through 10 a transverse link can beincluded to add a third degree of freedom, thus allowing the point 151to moved revolutely as well as vertically and horizontally.

FIGS. 12a and 12 b illustrate the compound 60 and simple (or single) 10joint configurations, both of which are constructed of tubes of equalradius and thickness, and both having the same overall length. Though acompound joint 60 is not limited to use of identical parts. The compound(split-tube) flexure 60 shown in FIG. 12a includes a first tube 70having a wall 72 and having a first end 74 and a second end 76. A length78 being defined from the first end 74 to the second end 76. The wall 72also having a lengthwise slit 80. The compound flexure 60 also includesa second tube 110 having a wall 112 and having a first end 114 and asecond end 116. A length 118 being defined from the first end 114 to thesecond end 116. The wall 112 also having a lengthwise slit 119. Thesecond tube 110 is co-linear with the first tube 70 such that the firstends 74 and 114 face each other. A first link 90 is attached to thefirst tube 70 and the second tube 110 nearer to the first ends 74 and114 of the tubes than the second ends 76 and 116 of the tubes. A secondlink 100 is attached to the first tube 70 nearer its second end 76 thanits first end 74. The second link 100 is also attached to the secondtube nearer its second end 116 than its first end 114.

In the embodiment shown in FIG. 12a, the second link 100 is a bifurcatedlink having first 122 and second 124 parallel portions attached to thefirst 70 and second 110 tubes, respectively. And the slits 80 and 119are co-linear. The first 90 and second 100 links are attached to thefirst 70 and second 110 tubes at locations lying on a line parallel tothe length of the tubes. While this is not required, it is one of thepreferred embodiments of the invention because the axis of rotation isconstant along the tubes opposite the slit. The axes of rotation of thetubes should be co-linear.

Mechanical analysis shows that for the revolute, axial, and bendingstiffnesses defined previously, the compound configuration exhibits 4times the revolute stiffness, 64 times the axial stiffness, and 16 timesthe bending stiffness of the simple joint configuration. A flexure-basedrevolute joint can be characterized by the ratio of revolute stiffnessto axial stiffness and the ratio of revolute stiffness to bendingstiffness, both of which in the ideal case would approach zero. Thoughthe absolute value of the revolute stiffness is 4 times larger for thecompound joint, the ratio of revolute to axial stiffness and of revoluteto bending stiffness are 16 and 4 times smaller, respectively, than thesimple joint.

It will be apparent to those skilled in the mechanical arts to combine anumber of single flexures as well as compound flexures to achieveminimal rotational force resistance about select axes of rotation whilemaintaining relatively large force resistance along other axes.

Thus, although there have been described particular embodiments of thepresent invention of a new and useful Split Tube Flexure, it is notintended that such references be construed as limitations upon the scopeof this invention except as set forth in the following claims.

What is claimed is:
 1. The split-tube flexure joint comprising a firsttube having a wall, a first end, and a second end, a length of the tubebeing defined from the first end to the second end, the wall having alengthwise slit defined therein along a longitudinal axis of the tubefrom the first end to the second end; first and second linkstangentially attached to an exterior surface of the wall at respectivefirst and second contact points along a contact line extending proximatethe exterior surface of the wall and parallel to the longitudinal axisof the tube; and, each of the first and second links extendingtransversely to the slit; the first tube having an axis of rotation; asecond tube having an axis of rotation, a first end, and a second end,the first end of the second tube facing the first end of the first tubesuch that the axis of rotation of each tube aligns; the first link beingattached to each of the first and second tube nearer to their first endsthan to their second ends; and the second link being attached to each ofthe first tube and second tubes nearer to their second ends than totheir first ends.
 2. The split-tube flexure joint of claim 1, whereinthe first and second links are attached to the second tube at locationslying on a line parallel to the length of the tubes.
 3. The split-tubeflexure joint of claim 1, wherein the first link and the second link areattached to the second tube at locations lying on a line parallel to thelength of the tubes.
 4. The split-tube flexure joint of claim 1, whereinthe first tube is co-linear with the second tube.
 5. The split-tubeflexure joint of claim 1, wherein the first and second links areattached to the first and second tubes at locations lying on a lineparallel to the length of the tubes.
 6. The split-tube flexure joint ofclaim 1, wherein the first and second links are parallel to each otherwhen the flexure-joint is in a relaxed state.
 7. The split-tube flexurejoint of claim 1, wherein the first and second links are transverse toeach other when the flexure-joint is in a relaxed state.
 8. A split-tubeflexure comprising: a first thin-walled hollow shaft having a wall, alength, the wall having a lengthwise slit defined therein; a first linktangentially attached to the wall; a second link tangentially attachedto the wall, the second link having a length, the second link lengthbeing transverse to slit; the first and second links contacting the wallat respective first and second contact points along a contact lineextending proximate an exterior surface of the wall and parallel to thelength of the shaft; the first shaft including first and second ends andan axis of rotation; a second thin-walled hollow shaft having a firstend, a second end, a lengthwise slit defined therein and an axis ofrotation, the first end facing the first end of the first shaft suchthat the axis of rotation of the first shaft is co-linear with the axisof rotation of the second shaft; the first link being attached to boththe first and second shafts nearer to their first ends than to theirsecond ends; and the second link being attached to both the first andsecond shafts nearer to their second ends than to their first ends. 9.The split-tube flexure of claim 8, wherein the second link comprises abifurcated link having first and second parallel link portions attachedto the first and second shafts, respectively.
 10. The split-tube flexureof claim 8, wherein the slit of each the first and the second shafts areco-linear.
 11. The split-tube flexure of claim 10, wherein the first andsecond links are attached to the first and second shafts at locationsdiametrically opposed to the slits in the first and second shafts. 12.The split-tube flexure of claim 8, wherein the first and second linksare attached to the first and second shafts at locations lying on a lineparallel to the first shaft.
 13. A method of providing pivotal motion,the method comprising constructing a first flexure joint by providing afirst thin-walled shaft having a lengthwise slit; attaching a first linkto the thin-walled shaft, the first link extending transversely to theslit; attaching a second link to the first thin-walled shaft; extendingthe second link transversely to the slit; the first and second linkstangentially contacting the shaft at respective first and second contactpoints along a contact line extending proximate an exterior surface ofthe shaft and parallel to the slit; applying a first force at the firstlink, the first force having a vector component perpendicular to theslit; and creating at the second link reactive forces reactive to thefirst force, and thereby achieving minimal force resistance along anaxis of rotation and relatively large force resistance along other axisof the flexure joint.
 14. The method of claim 13 further comprisingforming a first compound split-tube flexure by providing a secondthin-walled shaft having a lengthwise slit; aligning the second shaftco-linear with the first shaft so that a first end of the first shaftfaces a first end of the second shaft; attaching the first link to thefirst and second shafts proximate their first ends; and attaching thesecond link to the first and second shafts at locations spaced from thefirst link toward the second ends of each shaft.
 15. The method of claim14, further comprising attaching the first and second links to the firstand second shafts at locations lying on a line parallel to the firstshaft.
 16. The method of claim 14, further comprising forming a linkage,including the step of connecting a second compound split-tube flexure tothe first compound split-tube flexure.
 17. The method of claim 13,further comprising forming a linkage, including the step of constructinga second flexure joint similar to the first flexure joint and connectingthe second flexure joint to the first flexure joint.